Since we recover the results in [1]} when \(\rho =0\) , let us assume this and explain how our results differ from [1]} when they are applied to the CH equation. If \(\rho =0\)  and we fix a value of \(t\) , then the conditions mentioned in the last paragraph would necessarily imply that the function (REF ) vanishes provided that \((t,x)\)  belongs to such open set. As a result, the relation (REF ) tell us that \(f_t(x)\)  vanishes and for the same value of \(t\) , the function \(u(t,x)=0\) , \(x\in \mathbb {E}\) . The point is: And the other values of \(t\) ? Our approach use the conservation of energy of the solutions to extend such result for other values of \(t\) . For a fixed \(t\) , the inequality (REF ) implies that the energy of the solutions vanishes if and only if \(f_t(\cdot )\)  vanishes. If this happens, the invariance of the functional (REF ) implies that the solution is zero for each \(t\)  as long as the solution exists.
